On a hitting probability of Gaussian random vector into a small ball in a Hilbert space

Let $\xi$ be a Gaussian random vector taking its value in a Hilbert space $H$. Denote by $\theta(a,z)$ the ball in $H$ with center $a$ and radius $z$. Let $I(a,z)=\Prob\{\xi\in\theta(a,z)\}$, $z\to0$. We give some asymptotic formulas for $I(a,z)$ valid when $z\to0$.